(iv)
Let \(e\in [0,1)\)  satisfy \({\alpha }_k(e)<{\alpha }_s(e) \le {\alpha }_m(e)\) .
As the limiting case of the cases (iii) of Theorem REF ,
the matrix \(\xi _{{\alpha }_k(e),e}(2\pi )\)  must have Krein collision eigenvalues \({\sigma }(M) = \lbrace  {\lambda }_1,\bar{{\lambda }}_1, {\lambda }_2,\bar{{\lambda }}_2\rbrace \)  with \({\lambda }_1 = \bar{{\lambda }}_2 = e^{\sqrt{-1}{\theta }}\)  for some \({\theta }\in (0,\pi ) \cup (\pi ,2\pi )\) .
By Theorem REF  and the definition of \({\alpha }_k(e)\) ,
\(\pm 1\)  cannot be the eigenvalue of \(\xi _{{\alpha }_k(e),e}(2\pi )\) . Therefore, we must have that \(M \approx N_2({\omega },b)\)  for \({\omega }= e^{\sqrt{-1}{\theta }}\)  and some matrix
\(b=({\begin{matrix}b_1 & b_2\\ b_3 & b_4\end{matrix}})\) 
which has the form of (25-27) by Theorem 1.6.11 in p. 34 of [1]}.
Because \((I_2 \diamond -I_2)^{-1} N_2( e^{\sqrt{-1}{\theta }},b)(I_2 \diamond -I_2) = N_2( e^{\sqrt{-1}(2\pi -{\theta })},\hat{b}) \)  where
\(\hat{b}=({\begin{matrix}b_1 & -b_2\\ -b_3 & b_4\end{matrix}})\) .
We can always suppose \({\theta }\in (0,\pi )\)  without changing the fact \(M \approx N_2({\omega },b)\) .
